Subgroup ($H$) information
| Description: | $C_3:D_9$ |
| Order: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Index: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$b^{3}, c, d^{7}, d^{3}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_3^3.S_3^2$ |
| Order: | \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and supersolvable (hence solvable and monomial).
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4.C_3^4.C_2^3$, of order \(52488\)\(\medspace = 2^{3} \cdot 3^{8} \) |
| $\operatorname{Aut}(H)$ | $C_3^4.S_3^2$, of order \(2916\)\(\medspace = 2^{2} \cdot 3^{6} \) |
| $\operatorname{res}(S)$ | $C_3^4.S_3^2$, of order \(2916\)\(\medspace = 2^{2} \cdot 3^{6} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $C_3:D_9$, of order \(54\)\(\medspace = 2 \cdot 3^{3} \) |
Related subgroups
| Centralizer: | $C_2$ | ||||
| Normalizer: | $C_3:D_{18}$ | ||||
| Normal closure: | $C_3^2:D_9$ | ||||
| Core: | $C_3^2$ | ||||
| Minimal over-subgroups: | $C_3^2:D_9$ | $C_3:D_{18}$ | |||
| Maximal under-subgroups: | $C_3\times C_9$ | $C_3:S_3$ | $D_9$ | $D_9$ | $D_9$ |
Other information
| Number of subgroups in this conjugacy class | $9$ |
| Möbius function | $0$ |
| Projective image | $C_3^3.S_3^2$ |